Prediction (REF ) is only valid for free capillary–gravity waves, whereas dissipation rates of forced wave motions are generally more complex. A proper viscous WNL analysis would produce complex eigenmodes and responses (due to the phase shift owing to viscosity) and hence complex-valued normal form coefficients. For instance, among these coefficients, the imaginary part of \(\nu _{_{DC}}\)  (or \(\nu _{_{SC}}\) ) multiplied by \(|B|^2\)  in (REF ), could be interpreted as a sort of nonlinear damping [1]}, \(\left(\sigma +\text{Im}\left[\nu \right]|B|^2\right)\) , whose contribution to the overall dissipation mechanisms is expected to increase at larger wave amplitudes, hence influencing the location of the jump-down frequency. In contradistinction with the case of a pinned (or fixed) contact line, a formal viscous analysis undertaking the case of a moving contact line would require the introduction of a slip length model in order to regularize the well-known contact line stress-singularity [2]}, [3]}, [4]}, [5]}, [6]}.
